Overview
The latent variable approach is a mathematical trick which allows one to compute credit default statistics easily, especially when correlations among defaults are involved.
While strictly speaking, the latent variables introduced into the problem need not have an economic interpretation, they can be set within the framework of the merton model, where the latent variable is interpreted as the asset value of the firm, or a systemic risk variable that drives the asset values of the firms being considered.
Latent Variables
In order to simplify the calculations, modelers often model defaults using latent variables. That is, we assume there are other random variables to which the default variables are tied.
For example, assume that we have a set of Bernoulli distributed random variables {% y_i %} representing default of asset {% i %} in the portfolio. (That is {% y_1 = 1 %} for default, {% y_i = 0 %} for no default)
We then assume that there is a set of random variables {% A_i %} such that the following holds.
{% y_i = 1 %} when {% A_i \leq d_i %}
and
{% y_i = 0 %} when {% A_i > d_i %}
Then the probability of default is given by
{% prob_i = \Phi(d_i) %}
where {% \Phi %} is the
cumulative distribution function
for the random variable {% A_i %}.
This implies that
{% d_i = \Phi^{-1}(prob_i) %}
Probability of Joint Default
Then the joint probability of default can be modeled as
{% Prob(y_1 = 1, y_j = 1) = Prob(A_i \leq d_i \: and \: A_j \leq d_j) %}
Typically, the {% A_i %} variables are modeled as
{% A_i = w_i Z + \sqrt{1-w_i^2} e_i %}
where both {% Z %} and {% e_i %} are
normally distributed
random variables with mean {% 0 %} and variance equal to {% 1 %}, and with
{% cov(e_i, e_j) = 0 %}
and also
{% cov(e_i, Z) = 0 %}
{% A_i %} is then a
normal distribution
by
sums of normals.
The {% w_i %}
are then chosen in order to create correlations among the variables. That is, each
{% A_i %} is dependent on {% Z %}, and this dependence creates correlations among the
{% A_i %} and therefore, correlations among the defaults, {% y_i %}.
Calculating Probability of Joint Default
The correlation of the latent variables is given by
{% \rho_{ij}^{asset} = \frac{cov(A_i, A_j)}{\sigma_{Ai} \sigma_{Aj}} %}
see correlation
For the variables {% A_i %} defined above, we have
{% \rho_{ij}^{asset} = \frac{cov(w_i Z + \sqrt{1-w_i^2} e_i , w_j Z + \sqrt{1-w_j^2} e_j)}{1 \times 1} %}
Note, the variables were constructed such that the variances were equal to 1.
Because {% e_i %} and {% e_j %} are uncorrelated, we have
{% cov(w_i Z + \sqrt{1-w_i^2} e_i , w_j Z + \sqrt{1-w_j^2} e_j) = cov(w_i Z , w_j Z ) %}
which then implies that
{% \rho_{ij}^{asset} = w_i \times w_j %}
Given this, the probability can be calculated as
{% Prob(A_i \leq d_i , A_j \leq d_j) = prob_{ij} = \Phi_2(d_i,d_j, \rho_{ij}^{asset}) %}
where {% \Phi %} is the standard bivariate normal distribution function with the given correlation.
If {% prob_{ij} %}, {% prob_i %} and {% prob_j %} are known, then {% \rho_{ij}^{asset} %} can be found using such methods as Root Finding. That is we have
{% \Phi_2(d_i,d_j, \rho_{ij}^{asset}) - prob_{ij} = 0 %}
and we seek to find {% \rho_{ij}^{asset} %} that makes this equation true.